Complemented lattice

<h2 id="definition-and-basic-properties">Definition and basic properties</h2>
A complemented lattice is a bounded lattice (with <a href="/facts/Least_element/JVn2B3mo">least element</a> 0 and <a href="/facts/Greatest_element/JVn2B3mo">greatest element</a> 1), in which every element a has a complement, i.e. an element b such that

a ∨ b = 1     and    a ∧ b = 0.
In general an element may have more than one complement. However, in a (bounded) <a href="/facts/Distributive_lattice/97DFz1cu">distributive lattice</a> every element will have at most one complement.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> A lattice in which every element has exactly one complement is called a uniquely complemented lattice<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>
A lattice with the property that every interval (viewed as a sublattice) is complemented is called a relatively complemented lattice. In other words, a relatively complemented lattice is characterized by the property that for every element a in an interval [c, d] there is an element b such that

a ∨ b = d     and    a ∧ b = c.
Such an element b is called a complement of a relative to the interval. 
A distributive lattice is complemented if and only if it is bounded and relatively complemented.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a><a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> The lattice of <a href="/facts/Vector_subspace/2wtKTgHL">subspaces</a> of a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> provide an example of a complemented lattice that is not, in general, distributive.

<h2 id="orthocomplementation">Orthocomplementation</h2>
See also: <a href="/facts/De_Morgan_algebra/rLygz2hz">De Morgan algebra</a>

An orthocomplementation on a bounded lattice is a function that maps each element a to an "orthocomplement" a⊥ in such a way that the following axioms are satisfied:<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

Complement law
a⊥ ∨ a = 1 and a⊥ ∧ a = 0.
Involution law
a⊥⊥ = a.
Order-reversing
if a ≤ b then b⊥ ≤ a⊥.
An orthocomplemented lattice or ortholattice is a bounded lattice equipped with an orthocomplementation. The lattice of subspaces of an <a href="/facts/Inner_product_space/HoyjElEy">inner product space</a>, and the <a href="/facts/Orthogonal_complement/a13oQBsz">orthogonal complement</a> operation, provides an example of an orthocomplemented lattice that is not, in general, distributive.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

<a href="/facts/Boolean_algebra_(structure)/cHbo65jm">Boolean algebras</a> are a special case of orthocomplemented lattices, which in turn are a special case of complemented lattices (with extra structure). The ortholattices are most often used in <a href="/facts/Quantum_logic/yhgQIwyP">quantum logic</a>, where the <a href="/facts/Closed_set/upYnRTUj">closed</a> <a href="/facts/Linear_subspace/2wtKTgHL">subspaces</a> of a <a href="/facts/Separable_space/2bmU73bk">separable</a> <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> represent quantum propositions and behave as an orthocomplemented lattice.
Orthocomplemented lattices, like Boolean algebras, satisfy <a href="/facts/De_Morgan%2527s_laws/dj3F8xpt">de Morgan's laws</a>:

<ul><li>(a ∨ b)⊥ = a⊥ ∧ b⊥</li>
<li>(a ∧ b)⊥ = a⊥ ∨ b⊥.</li></ul>
<h2 id="orthomodular-lattices">Orthomodular lattices</h2>
A lattice is called <a href="/facts/Modular_lattice/CLe2aDq7">modular</a> if for all elements a, b and c the implication

if a ≤ c, then a ∨ (b ∧ c) = (a ∨ b) ∧ c
holds. This is weaker than <a href="/facts/Distributive_lattice/97DFz1cu">distributivity</a>; e.g. the above-shown lattice M3 is modular, but not distributive. 
A natural further weakening of this condition for orthocomplemented lattices, necessary for applications in quantum logic, is to require it only in the special case b = a⊥. An orthomodular lattice is therefore defined as an orthocomplemented lattice such that for any two elements the implication

if a ≤ c, then a ∨ (a⊥ ∧ c) = c
holds.
Lattices of this form are of crucial importance for the study of <a href="/facts/Quantum_logic/yhgQIwyP">quantum logic</a>, since they are part of the axiomisation of the <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> <a href="/facts/Mathematical_formulation_of_quantum_mechanics/fRLw4sFU">formulation</a> of <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>. <a href="/facts/Garrett_Birkhoff/5jeI7Wmr">Garrett Birkhoff</a> and <a href="/facts/John_von_Neumann/aUk6urof">John von Neumann</a> observed that the <a href="/facts/Propositional_logic/MNt5v31V">propositional</a> <a href="/facts/Logical_calculus/KaeDAjK7">calculus</a> in quantum logic is "formally indistinguishable from the calculus of linear subspaces [of a Hilbert space] with respect to <a href="/facts/Intersection/hVW6QFWD">set products</a>, <a href="/facts/Linear_subspace/2wtKTgHL">linear sums</a> and orthogonal complements" corresponding to the roles of and, or and not in Boolean lattices. This remark has spurred interest in the closed subspaces of a Hilbert space, which form an orthomodular lattice.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Pseudocomplemented_lattice/q3mSbcIY">Pseudocomplemented lattice</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Birkhoff, Garrett (1961). Lattice Theory. American Mathematical Society.</li>
<li><a href="/facts/George_Gr%25C3%25A4tzer/RtSM5JdW">Grätzer, George</a> (1971). Lattice Theory: First Concepts and Distributive Lattices. W. H. Freeman and Company. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-7167-0442-3.</li>
<li>Grätzer, George (1978). General Lattice Theory. Basel, Switzerland: Birkhäuser. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-12-295750-5.</li>
<li>Rutherford, Daniel Edwin (1965). Introduction to Lattice Theory. Oliver and Boyd.</li></ul>
<h2 id="external-links">External links</h2>
<table><tbody><tr><td></td></tr></tbody></table>
<ul><li><a href="https://planetmath.org/ComplementedLattice">"Complemented lattice"</a>. <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>.</li>
<li><a href="https://planetmath.org/RelativeComplement">"Relative complement"</a>. <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>.</li>
<li><a href="https://planetmath.org/UniquelyComplementedLattice">"Uniquely complemented lattice"</a>. <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>.</li>
<li><a href="https://planetmath.org/OrthocomplementedLattice">"Orthocomplemented lattice"</a>. <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Grätzer (1971), Lemma I.6.1, p. 47. Rutherford (1965), Theorem 9.3 p. 25. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Stern, Manfred (1999), Semimodular Lattices: Theory and Applications, Encyclopedia of Mathematics and its Applications, Cambridge University Press, p. 29, ISBN 9780521461054. <a href="9780521461054" target="_blank">9780521461054</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Grätzer (1971), Lemma I.6.2, p. 48. This result holds more generally for modular lattices, see Exercise 4, p. 50. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Birkhoff (1961), Corollary IX.1, p. 134 <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Stern (1999), p. 11. - Stern, Manfred (1999), Semimodular Lattices: Theory and Applications, Encyclopedia of Mathematics and its Applications, Cambridge University Press, p. 29, ISBN 9780521461054 <a href="https://books.google.com/books?id=VVYd2sC19ogC&pg=PA29" target="_blank">https://books.google.com/books?id=VVYd2sC19ogC&pg=PA29</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">The Unapologetic Mathematician: Orthogonal Complements and the Lattice of Subspaces. <a href="http://unapologetic.wordpress.com/2009/05/07/orthogonal-complements-and-the-lattice-of-subspaces/" target="_blank">http://unapologetic.wordpress.com/2009/05/07/orthogonal-complements-and-the-lattice-of-subspaces/</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Ranganathan Padmanabhan; Sergiu Rudeanu (2008). Axioms for lattices and boolean algebras. World Scientific. p. 128. ISBN 978-981-283-454-6. <a href="978-981-283-454-6" target="_blank">978-981-283-454-6</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Complemented lattice open-in-new

Complemented lattice